Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 095, 37 pages      arXiv:1207.4386      https://doi.org/10.3842/SIGMA.2012.095

Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles

Andrey M. Levin a, b, Mikhail A. Olshanetsky b, Andrey V. Smirnov b, c and Andrei V. Zotov b
a) Laboratory of Algebraic Geometry, GU-HSE, 7 Vavilova Str., Moscow, 117312, Russia
b) Institute of Theoretical and Experimental Physics, Moscow, 117218, Russia
c) Department of Mathematics, Columbia University, New York, NY 10027, USA

Received July 14, 2012, in final form November 29, 2012; Published online December 10, 2012

Abstract
We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint $G$-bundles of different topological types over complex curves $\Sigma_{g,n}$ of genus $g$ with $n$ marked points. The bundles are defined by their characteristic classes - elements of $H^2(\Sigma_{g,n},\mathcal{Z}(G))$, where $\mathcal{Z}(G)$ is a center of the simple complex Lie group $G$. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the underlying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly for elliptic curves with marked points and prove its flatness.

Key words: integrable system; KZB equation; Hitchin system; characteristic class.

pdf (648 kb)   tex (48 kb)

References

  1. Aminov G., Arthamonov S., Reduction of the elliptic ${\rm SL}(N,{\mathbb C})$ top, J. Phys. A: Math. Theor. 44 (2011), 075201, 34 pages, arXiv:1009.1867.
  2. Aminov G., Arthamonov S., Levin A.M., Olshanetsky M.A., Zotov A.V., Around Painlevé VI field theory, submitted.
  3. Atiyah M.F., Vector bundles over an elliptic curve, Proc. London Math. Soc. 7 (1957), 414-452.
  4. Axelrod S., Pietra S.D., Witten E., Geometric quantization of Chern-Simons gauge theory, J. Differential Geom. 33 (1991), 787-902.
  5. Beauville A., Conformal blocks, fusion rules and the Verlinde formula, in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., Vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, 75-96, alg-geom/9405001.
  6. Beauville A., Laszlo Y., Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385-419, alg-geom/9309003.
  7. Ben-Zvi D., Frenkel E., Geometric realization of the Segal-Sugawara construction, in Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., Vol. 308, Cambridge University Press, Cambridge, 2004, 46-97, math.AG/0301206.
  8. Bernard D., On the Wess-Zumino-Witten models on Riemann surfaces, Nuclear Phys. B 309 (1988), 145-174.
  9. Bernard D., On the Wess-Zumino-Witten models on the torus, Nuclear Phys. B 303 (1988), 77-93.
  10. Bernstein J., Schwarzman O., Chevalley's theorem for complex crystallographic Coxeter groups, Funct. Anal. Appl. 12 (1978), 308-310.
  11. Bernstein J., Schwarzman O., Complex crystallographic Coxeter groups and affine root systems, J. Nonlinear Math. Phys. 13 (2006), 163-182.
  12. Bourbaki N., Lie groups and Lie algebras, Chapters 4-6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002.
  13. Braden H.W., Dolgushev V.A., Olshanetsky M.A., Zotov A.V., Classical $r$-matrices and the Feigin-Odesskii algebra via Hamiltonian and Poisson reductions, J. Phys. A: Math. Gen. 36 (2003), 6979-7000, hep-th/0301121.
  14. Bulycheva K., Monopole solutions to the Bogomolny equation as three-dimensional generalizations of the Kronecker series, Theoret. and Math. Phys. 172 (2012), 1232-1242, arXiv:1203.4674.
  15. Chernyakov Yu.B., Levin A.M., Olshanetsky M.A., Zotov A.V., Elliptic Schlesinger system and Painlevé VI, J. Phys. A: Math. Gen. 39 (2006), 12083-12101, nlin.SI/0602043.
  16. Enriquez B., Rubtsov V., Hecke-Tyurin parametrization of the Hitchin and KZB systems, in Moscow Seminar on Mathematical Physics. II, Amer. Math. Soc. Transl. Ser. 2, Vol. 221, Amer. Math. Soc., Providence, RI, 2007, 1-31, math.AG/9911087.
  17. Etingof P., Schiffmann O., Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical $R$-matrices corresponding to generalized Belavin-Drinfeld triples, Math. Res. Lett. 6 (1999), 593-612, math.QA/9908115.
  18. Etingof P., Schiffmann O., Varchenko A., Traces of intertwiners for quantum groups and difference equations, Lett. Math. Phys. 62 (2002), 143-158, math.QA/0207157.
  19. Etingof P., Varchenko A., Geometry and classification of solutions of the classical dynamical Yang-Baxter equation, Comm. Math. Phys. 192 (1998), 77-120, q-alg/9703040.
  20. Faltings G., A proof for the Verlinde formula, J. Algebraic Geom. 3 (1994), 347-374.
  21. Fehér L., Pusztai B.G., Generalizations of Felder's elliptic dynamical $r$-matrices associated with twisted loop algebras of self-dual Lie algebras, Nuclear Phys. B 621 (2002), 622-642, math.QA/0109132.
  22. Felder G., The KZB equations on Riemann surfaces, in Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 687-725, hep-th/9609153.
  23. Felder G., Gawedzki K., Kupiainen A., Spectra of Wess-Zumino-Witten models with arbitrary simple groups, Comm. Math. Phys. 117 (1988), 127-158.
  24. Felder G., Wieczerkowski C., Conformal blocks on elliptic curves and the Knizhnik-Zamolodchikov-Bernard equations, Comm. Math. Phys. 176 (1996), 133-161, hep-th/9411004.
  25. Frenkel E., Lectures on the Langlands program and conformal field theory, in Frontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin, 2007, 387-533, hep-th/0512172.
  26. Friedman R., Morgan J.W., Holomorphic principal bundles over elliptic curves, math.AG/9811130.
  27. Friedman R., Morgan J.W., Holomorphic principal bundles over elliptic curves. II. The parabolic construction, J. Differential Geom. 56 (2000), 301-379, math.AG/0006174.
  28. Friedman R., Morgan J.W., Witten E., Principal $G$-bundles over elliptic curves, Math. Res. Lett. 5 (1998), 97-118, alg-geom/9707004.
  29. Fuchs J., Schweigert C., The action of outer automorphisms on bundles of chiral blocks, Comm. Math. Phys. 206 (1999), 691-736, hep-th/9805026.
  30. Gorsky A., Nekrasov N., Hamiltonian systems of Calogero-type, and two-dimensional Yang-Mills theory, Nuclear Phys. B 414 (1994), 213-238, hep-th/9304047.
  31. Gukov S., Witten E., Branes and quantization, Adv. Theor. Math. Phys. 13 (2009), 1445-1518, arXiv:0809.0305.
  32. Harnad J., Quantum isomonodromic deformations and the Knizhnik-Zamolodchikov equations, in Symmetries and Integrability of Difference Equations (Estérel, PQ, 1994), CRM Proc. Lecture Notes, Vol. 9, Amer. Math. Soc., Providence, RI, 1996, 155-161, hep-th/9406078.
  33. Hitchin N.J., Flat connections and geometric quantization, Comm. Math. Phys. 131 (1990), 347-380.
  34. Hitchin N.J., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114.
  35. Hori K., Global aspects of gauged Wess-Zumino-Witten models, Comm. Math. Phys. 182 (1996), 1-32, hep-th/9411134.
  36. Ivanov D.A., Knizhnik-Zamolodchikov-Bernard equations on Riemann surfaces, Internat. J. Modern Phys. A 10 (1995), 2507-2536, hep-th/9410091.
  37. Ivanova T.A., Lechtenfeld O., Popov A.D., Rahn T., Instantons and Yang-Mills flows on coset spaces, Lett. Math. Phys. 89 (2009), 231-247, arXiv:0904.0654.
  38. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  39. Kapustin A., Witten E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1-236, hep-th/0604151.
  40. Knizhnik V.G., Zamolodchikov A.B., Current algebra and Wess-Zumino model in two dimensions, Nuclear Phys. B 247 (1984), 83-103.
  41. Korotkin D., Samtleben H., On the quantization of isomonodromic deformations on the torus, Internat. J. Modern Phys. A 12 (1997), 2013-2029, hep-th/9511087.
  42. Kuroki G., Takebe T., Twisted Wess-Zumino-Witten models on elliptic curves, Comm. Math. Phys. 190 (1997), 1-56, q-alg/9612033.
  43. Levin A.M., Olshanetsky M.A., Double coset construction of moduli space of holomorphic bundles and Hitchin systems, Comm. Math. Phys. 188 (1997), 449-466, alg-geom/9605005.
  44. Levin A.M., Olshanetsky M.A., Hierarchies of isomonodromic deformations and Hitchin systems, in Moscow Seminar in Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 191, Amer. Math. Soc., Providence, RI, 1999, 223-262.
  45. Levin A.M., Olshanetsky M.A., Smirnov A.V., Zotov A.V., Calogero-Moser systems for simple Lie groups and characteristic classes of bundles, J. Geom. Phys. 62 (2012), 1810-1850, arXiv:1007.4127.
  46. Levin A.M., Olshanetsky M.A., Smirnov A.V., Zotov A.V., Characteristic classes and Hitchin systems. General construction, Comm. Math. Phys. 316 (2012), 1-44, arXiv:1006.0702.
  47. Levin A.M., Olshanetsky M.A., Smirnov A.V., Zotov A.V., Characteristic classes of ${\rm SL}(N)$-bundles and quantum dynamical elliptic $R$-matrices, J. Phys. A: Math. Theor., to appear, arXiv:1208.5750.
  48. Levin A.M., Olshanetsky M.A., Zotov A.V., Hitchin systems - symplectic Hecke correspondence and two-dimensional version, Comm. Math. Phys. 236 (2003), 93-133, nlin.SI/0110045.
  49. Levin A.M., Olshanetsky M.A., Zotov A.V., Monopoles and modifications of bundles over elliptic curves, SIGMA 5 (2009), 065, 22 pages, arXiv:0811.3056.
  50. Levin A.M., Olshanetsky M.A., Zotov A.V., Painlevé VI, rigid tops and reflection equation, Comm. Math. Phys. 268 (2006), 67-103, math.QA/0508058.
  51. Levin A.M., Zotov A.V., On rational and elliptic forms of Painlevé VI equation, in Moscow Seminar on Mathematical Physics. II, Amer. Math. Soc. Transl. Ser. 2, Vol. 221, Amer. Math. Soc., Providence, RI, 2007, 173-183.
  52. Looijenga E., Root systems and elliptic curves, Invent. Math. 38 (1976), 17-32.
  53. Mironov A., Morozov A., Runov B., Zenkevich Y., Zotov A., Spectral duality between Heisenberg chain and Gaudin model, Lett. Math. Phys., to appear, arXiv:1206.6349.
  54. Mironov A., Morosov A., Shakirov Sh., Towards a proof of AGT conjecture by methods of matrix models, Internat. J. Modern Phys. A 27 (2012), 1230001, 32 pages, arXiv:1011.5629.
  55. Mironov A., Morozov A., Zenkevich Y., Zotov A., Spectral duality in integrable systems from AGT conjecture, JETP Lett., to appear, arXiv:1204.0913.
  56. Narasimhan M.S., Seshadri C.S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540-567.
  57. Nekrasov N., Holomorphic bundles and many-body systems, Comm. Math. Phys. 180 (1996), 587-603, hep-th/9503157.
  58. Nekrasov N., Pestun V., Seiberg-Witten geometry of four dimensional $N=2$ quiver gauge theories, arXiv:1211.2240.
  59. Olshanetsky M.A., Three lectures on classical integrable systems and gauge field theories, Phys. Part. Nuclei 40 (2009), 93-114, arXiv:0802.3857.
  60. Olshanetsky M.A., Zotov A. V., Isomonodromic problems on elliptic curve, rigid tops and reflection equations, in Elliptic Integrable Systems, Rokko Lectures in Mathematics, Vol. 18, Kobe University, 2005, 149-172.
  61. Presley A., Segal G., Loop groups, Clarendon Press, Oxford, 1986.
  62. Reshetikhin N., The Knizhnik-Zamolodchikov system as a deformation of the isomonodromy problem, Lett. Math. Phys. 26 (1992), 167-177.
  63. Schweigert C., On moduli spaces of flat connections with non-simply connected structure group, Nuclear Phys. B 492 (1997), 743-755, hep-th/9611092.
  64. Simpson C.T., Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), 713-770.
  65. Smirnov A.V., Integrable ${\rm sl}(N,{\mathbb C})$-tops as Calogero-Moser systems, Theoret. and Math. Phys. 158 (2009), 300-312, arXiv:0809.2187.
  66. Takasaki K., Gaudin model, KZ equation and an isomonodromic problem on the torus, Lett. Math. Phys. 44 (1998), 143-156, hep-th/9711058.
  67. Zabrodin A.V., Zotov A.V., Quantum Painlevé-Calogero correspondence, J. Math. Phys. 53 (2012), 073507, 19 pages, arXiv:1107.5672.
  68. Zabrodin A.V., Zotov A.V., Quantum Painlevé-Calogero correspondence for Painlevé VI, J. Math. Phys. 53 (2012), 073508, 19 pages, arXiv:1107.5672.
  69. Zotov A.V., 1+1 Gaudin model, SIGMA 7 (2011), 067, 26 pages, arXiv:1012.1072.
  70. Zotov A.V., Classical integrable systems and their field-theoretical generalizations, Phys. Part. Nuclei 37 (2006), 759-842.
  71. Zotov A.V., Elliptic linear problem for the Calogero-Inozemtsev model and Painlevé VI equation, Lett. Math. Phys. 67 (2004), 153-165, hep-th/0310260.
  72. Zotov A.V., Chernyakov Yu.B., Integrable multiparticle systems obtained by means of the Inozemtsev limit, Theoret. and Math. Phys. 129 (2001), 1526-1542, hep-th/0102069.
  73. Zotov A.V., Levin A.M., An integrable system of interacting elliptic tops, Theoret. and Math. Phys. 146 (2006), 45-52.
  74. Zotov A.V., Levin A.M., Olshanetsky M.A., Chernyakov Yu.B., Quadratic algebras associated with elliptic curves, Theoret. and Math. Phys. 156 (2008), 1103-1122, arXiv:0710.1072.

Previous article  Next article   Contents of Volume 8 (2012)